So is it the physical form of the brain that gives it that capacity, or is it simply axiomatic that humans are different from other animals?
I'm not sure what the physical difference would be, though I suspect there is a difference (possibly in frequency calibration) that allows the human to transmit/receive the neshama as well as the nephesh.
Are apes having sign language conscious?
I don't believe so. Apes learning sign language is much like mice learning mazes or dogs learning tricks. If any animal shows willfulness, to me it would be cats. (LOL!) The others display a learned response.
Is there some test, even a hypothetical one, that could settle the issue?
Again, I am not certain. I suspect the Penrose/Hameroff model is the best one currently proposed to broach the subject (the OR theory.) But there are others, and a great website to track the development of hypotheses is PSYCHE An Interdisciplinary Journal of Research on Consciousness
I agree on cats, but I assume you've never owned hounds. Or apes.
I'm to have to agree to disagree with you on the privileged nature of humans. My own experience with animals and with retarded humans leads me to believe firmly that brain structure is destiny. Just as I see a (usually) gradual sloping off of our ability to see the future, I see a gradual sloping off of consciousness, consistent with brain size, structure and health. Language does appear to be a giant step, perhaps one that looks like a cliff to you, but I'm betting it will be accounted for by brain structure -- something that we can see, study, and eventually emulate in A.I.